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\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{6}   &  -7.0 & + 10.00 x_{1} & -8.00 x_{2} & +  5.00 x_{3} & +  1.00 x_{4} & -5.00 x_{5}\\
 x_{7}   &  48.0 & +  2.00 x_{1} & -10.00 x_{2} & -9.00 x_{3} & -6.00 x_{4} & +  5.00 x_{5}\\
 x_{8}   &  2.0 & -10.00 x_{1} & +  2.00 x_{2} & -3.00 x_{3} & + 10.00 x_{4} & -3.00 x_{5}\\
 x_{9}   &  -1.0 & +  8.00 x_{1} & -9.00 x_{2} & -7.00 x_{3} & +  7.00 x_{4} & -3.00 x_{5}\\
 x_{10}   &  -7.0 & +  7.00 x_{1} & -3.00 x_{2} & -3.00 x_{3} & +  9.00 x_{4} & -4.00 x_{5}\\
 x_{11}   &  34.0 & -10.00 x_{1} & -3.00 x_{2} & +  3.00 x_{3} & -7.00 x_{4} & +  5.00 x_{5}\\
\hline
z    &  0.0 & -4.00 x_{1} & +  4.00 x_{2} & -3.00 x_{3} & -5.00 x_{4} & +  5.00 x_{5}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{5}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,11$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & -10.00 y_{6} & -2.00 y_{7} & + 10.00 y_{8} & -8.00 y_{9} & -7.00 y_{10} & + 10.00 y_{11}\\
 y_{2}   &  1.0 & +  8.00 y_{6} & + 10.00 y_{7} & -2.00 y_{8} & +  9.00 y_{9} & +  3.00 y_{10} & +  3.00 y_{11}\\
 y_{3}   &  1.0 & -5.00 y_{6} & +  9.00 y_{7} & +  3.00 y_{8} & +  7.00 y_{9} & +  3.00 y_{10} & -3.00 y_{11}\\
 y_{4}   &  1.0 & -1.00 y_{6} & +  6.00 y_{7} & -10.00 y_{8} & -7.00 y_{9} & -9.00 y_{10} & +  7.00 y_{11}\\
 y_{5}   &  1.0 & +  5.00 y_{6} & -5.00 y_{7} & +  3.00 y_{8} & +  3.00 y_{9} & +  4.00 y_{10} & -5.00 y_{11}\\
\hline
z    &  -0 & +  7.00 y_{6} & -48.00 y_{7} & -2.00 y_{8} & +  1.00 y_{9} & +  7.00 y_{10} & -34.00 y_{11}\\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 3 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{6}   &  0.181818181818 & -0.09 y_{1} & +  0.36 y_{7} & -0.09 y_{4} & -1.36 y_{9} & -1.45 y_{10} & +  1.55 y_{11}\\
 y_{2}   &  2.29090909091 & -0.75 y_{1} & + 11.78 y_{7} & -0.55 y_{4} & -0.78 y_{9} & -7.13 y_{10} & + 14.27 y_{11}\\
 y_{3}   &  0.336363636364 & +  0.48 y_{1} & +  8.87 y_{7} & +  0.18 y_{4} & + 12.13 y_{9} & +  8.01 y_{10} & -9.09 y_{11}\\
 y_{8}   &  0.0818181818182 & +  0.01 y_{1} & +  0.56 y_{7} & -0.09 y_{4} & -0.56 y_{9} & -0.75 y_{10} & +  0.55 y_{11}\\
 y_{5}   &  2.15454545455 & -0.43 y_{1} & -1.49 y_{7} & -0.73 y_{4} & -5.51 y_{9} & -5.54 y_{10} & +  4.36 y_{11}\\
\hline
z    &  1.10909090909 & -0.65 y_{1} & -46.58 y_{7} & -0.45 y_{4} & -7.42 y_{9} & -1.67 y_{10} & -24.27 y_{11}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.654545454545 & +  0.09 x_{6} & +  0.75 x_{2} & -0.48 x_{3} & -0.01 x_{8} & +  0.43 x_{5}\\
 x_{7}   &  46.5818181818 & -0.36 x_{6} & -11.78 x_{2} & -8.87 x_{3} & -0.56 x_{8} & +  1.49 x_{5}\\
 x_{4}   &  0.454545454545 & +  0.09 x_{6} & +  0.55 x_{2} & -0.18 x_{3} & +  0.09 x_{8} & +  0.73 x_{5}\\
 x_{9}   &  7.41818181818 & +  1.36 x_{6} & +  0.78 x_{2} & -12.13 x_{3} & +  0.56 x_{8} & +  5.51 x_{5}\\
 x_{10}   &  1.67272727273 & +  1.45 x_{6} & +  7.13 x_{2} & -8.01 x_{3} & +  0.75 x_{8} & +  5.54 x_{5}\\
 x_{11}   &  24.2727272727 & -1.55 x_{6} & -14.27 x_{2} & +  9.09 x_{3} & -0.55 x_{8} & -4.36 x_{5}\\
\hline
z    &  -1.10909090909 & -0.18 x_{6} & -2.29 x_{2} & -0.34 x_{3} & -0.08 x_{8} & -2.15 x_{5}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.654545454545 & +  0.09 x_{6} & +  0.75 x_{2} & -0.48 x_{3} & -0.01 x_{8} & +  0.43 x_{5}\\
 x_{7}   &  46.5818181818 & -0.36 x_{6} & -11.78 x_{2} & -8.87 x_{3} & -0.56 x_{8} & +  1.49 x_{5}\\
 x_{4}   &  0.454545454545 & +  0.09 x_{6} & +  0.55 x_{2} & -0.18 x_{3} & +  0.09 x_{8} & +  0.73 x_{5}\\
 x_{9}   &  7.41818181818 & +  1.36 x_{6} & +  0.78 x_{2} & -12.13 x_{3} & +  0.56 x_{8} & +  5.51 x_{5}\\
 x_{10}   &  1.67272727273 & +  1.45 x_{6} & +  7.13 x_{2} & -8.01 x_{3} & +  0.75 x_{8} & +  5.54 x_{5}\\
 x_{11}   &  24.2727272727 & -1.55 x_{6} & -14.27 x_{2} & +  9.09 x_{3} & -0.55 x_{8} & -4.36 x_{5}\\
\hline
z    &  -4.89090909091 & -0.82 x_{6} & -1.71 x_{2} & -0.16 x_{3} & -0.42 x_{8} & -0.35 x_{5}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{1}   &  0.654545454545 & +  0.09 x_{6} & +  0.75 x_{2} & -0.48 x_{3} & -0.01 x_{8} & +  0.43 x_{5}\\
 x_{7}   &  46.5818181818 & -0.36 x_{6} & -11.78 x_{2} & -8.87 x_{3} & -0.56 x_{8} & +  1.49 x_{5}\\
 x_{4}   &  0.454545454545 & +  0.09 x_{6} & +  0.55 x_{2} & -0.18 x_{3} & +  0.09 x_{8} & +  0.73 x_{5}\\
 x_{9}   &  7.41818181818 & +  1.36 x_{6} & +  0.78 x_{2} & -12.13 x_{3} & +  0.56 x_{8} & +  5.51 x_{5}\\
 x_{10}   &  1.67272727273 & +  1.45 x_{6} & +  7.13 x_{2} & -8.01 x_{3} & +  0.75 x_{8} & +  5.54 x_{5}\\
 x_{11}   &  24.2727272727 & -1.55 x_{6} & -14.27 x_{2} & +  9.09 x_{3} & -0.55 x_{8} & -4.36 x_{5}\\
\hline
z    &  -4.89090909091 & -0.82 x_{6} & -1.71 x_{2} & -0.16 x_{3} & -0.42 x_{8} & -0.35 x_{5}\\
\end{array}\]
Final Dictionary
Solution:  -4.89090909091
Num Pivots:  1
\end{document}
